Rational invariants for subgroups of S₅ and S₇

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\ldots,x_n)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any $\sigma\in G$, any $1\le i\le n$. Theorem. If $n\le 5$, then the fixed field $k(x_1,\ldots,x_n)^G$ is purely transcendental over $k$. We will show that $\bm{C}(x_1,\ldots,x_7)^G$ is also purely transcendental over $\bm{C}$ if $G$ is any transitive subgroups of $S_7$ other than $A_7$; a similar result is valid for solvable transitive subgroups of $S_{11}$.